# Electrical Impedance - Complex Voltage and Current

Complex Voltage and Current

In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as and .

begin{align} V &= |V|e^{j(omega t + phi_V)} \ I &= |I|e^{j(omega t + phi_I)} end{align}

Impedance is defined as the ratio of these quantities.

Substituting these into Ohm's law we have

begin{align} |V| e^{j(omega t + phi_V)} &= |I| e^{j(omega t + phi_I)} |Z| e^{jtheta} \ &= |I| |Z| e^{j(omega t + phi_I + theta)} end{align}

Noting that this must hold for all, we may equate the magnitudes and phases to obtain

begin{align} |V| &= |I| |Z| \ phi_V &= phi_I + theta end{align}

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.